Exploring black hole spacetimes with computers ´ Eric Gourgoulhon Laboratoire Univers et Th´ eories (LUTH) CNRS / Observatoire de Paris / Universit´ e Paris Diderot 92190 Meudon, France http://luth.obspm.fr/ ~ luthier/gourgoulhon/ based on a collaboration with Marek Abramowicz, Micha l Bejger, Philippe Grandcl´ ement, J´ erˆ ome Novak, Thibaut Paumard, Guy Perrin, Claire Som´ e, Odele Straub, Fr´ ed´ eric Vincent Riemann, Einstein and geometry 94th Encounter between Mathematicians and Theoretical Physicists Institut de Recherche Math´ ematique Avanc´ ee, Strasbourg 18-20 September 2014 ´ Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 1 / 62
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Exploring black hole spacetimes with computers
Eric Gourgoulhon
Laboratoire Univers et Theories (LUTH)CNRS / Observatoire de Paris / Universite Paris Diderot
92190 Meudon, France
http://luth.obspm.fr/~luthier/gourgoulhon/
based on a collaboration with
Marek Abramowicz, Micha l Bejger, Philippe Grandclement, Jerome Novak,Thibaut Paumard, Guy Perrin, Claire Some, Odele Straub, Frederic Vincent
Riemann, Einstein and geometry94th Encounter between Mathematicians and Theoretical Physicists
Institut de Recherche Mathematique Avancee, Strasbourg18-20 September 2014
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 1 / 62
Astrophysical motivation: we are about to see black holes!
The “no-hair” theorem
Dorochkevich, Novikov & Zel’dovich (1965), Israel (1967), Carter (1971),Hawking (1972)
Within 4-dimensional general relativity, a stationary black hole in an otherwiseempty universe is necessarily a Kerr-Newman black hole, which is a vacuumsolution of Einstein equation described by only three parameters:
the total mass M
the total angular momentum J
the total electric charge Q
=⇒ “a black hole has no hair” (John A. Wheeler)
Astrophysical black holes have to be electrically neutral:
Q = 0 : Kerr solution (1963)
Other special cases:
Q = 0 and a = 0 : Schwarzschild solution (1916)
a = 0: Reisnerr-Nordstrom solution (1916, 1918)
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 10 / 62
Astrophysical motivation: we are about to see black holes!
The “no-hair” theorem
Dorochkevich, Novikov & Zel’dovich (1965), Israel (1967), Carter (1971),Hawking (1972)
Within 4-dimensional general relativity, a stationary black hole in an otherwiseempty universe is necessarily a Kerr-Newman black hole, which is a vacuumsolution of Einstein equation described by only three parameters:
the total mass M
the total angular momentum J
the total electric charge Q
=⇒ “a black hole has no hair” (John A. Wheeler)
Astrophysical black holes have to be electrically neutral:
Q = 0 : Kerr solution (1963)
Other special cases:
Q = 0 and a = 0 : Schwarzschild solution (1916)
a = 0: Reisnerr-Nordstrom solution (1916, 1918)
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 10 / 62
Astrophysical motivation: we are about to see black holes!
Lowest order no-hair theorem: quadrupole moment
Asymptotic expansion (large r) of the metric in terms of multipole moments(Mk,Jk)k∈N [Geroch (1970), Hansen (1974)]:
Mk: mass 2k-pole moment
Jk: angular momentum 2k-pole moment
=⇒ For the Kerr metric, all the multipole moments are determined by (M,a):
M0 = M
J1 = aM = J/c
M2 = −a2M = − J2
c2M(∗) ← mass quadrupole moment
J3 = −a3MM4 = a4M
· · ·
Measuring the three quantities M , J , M2 provides a compatibility test w.r.t. theKerr metric, by checking (∗)
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 11 / 62
Astrophysical motivation: we are about to see black holes!
Lowest order no-hair theorem: quadrupole moment
Asymptotic expansion (large r) of the metric in terms of multipole moments(Mk,Jk)k∈N [Geroch (1970), Hansen (1974)]:
Mk: mass 2k-pole moment
Jk: angular momentum 2k-pole moment
=⇒ For the Kerr metric, all the multipole moments are determined by (M,a):
M0 = M
J1 = aM = J/c
M2 = −a2M = − J2
c2M(∗) ← mass quadrupole moment
J3 = −a3MM4 = a4M
· · ·
Measuring the three quantities M , J , M2 provides a compatibility test w.r.t. theKerr metric, by checking (∗)
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 11 / 62
Astrophysical motivation: we are about to see black holes!
Theoretical alternatives to the Kerr black hole
Within general relativity
The compact object is not a black hole but
a boson star
a gravastar
a dark star
...
Beyond general relativity
The compact object is a black hole but in a theory that differs from GR:
Einstein-Gauss-Bonnet with dilaton
Chern-Simons gravity
Horava-Lifshitz gravity
Einstein-Yang-Mills
...
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 12 / 62
Exploring spacetimes via numerical computations: the geodesic code GYOTO
Outline
1 Astrophysical motivation: we are about to see black holes!
2 Exploring spacetimes via numerical computations: the geodesic code GYOTO
3 Exploring spacetimes via symbolic computations: the SageManifolds project
4 Conclusion and perspectives
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 13 / 62
Exploring spacetimes via numerical computations: the geodesic code GYOTO
How to test the alternatives to the Kerr black hole?
Search for
stellar orbits deviating from Kerr timelike geodesics (GRAVITY)
accretion disk spectra different from those arising in Kerr metric (X-rayobservatories)
images of the black hole shadow different from that of a Kerr black hole(EHT)
Need for a good and versatile geodesic integratorto compute timelike geodesics (orbits) and null geodesics (ray-tracing) in any kind
of metric
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 14 / 62
Exploring spacetimes via numerical computations: the geodesic code GYOTO
How to test the alternatives to the Kerr black hole?
Search for
stellar orbits deviating from Kerr timelike geodesics (GRAVITY)
accretion disk spectra different from those arising in Kerr metric (X-rayobservatories)
images of the black hole shadow different from that of a Kerr black hole(EHT)
Need for a good and versatile geodesic integratorto compute timelike geodesics (orbits) and null geodesics (ray-tracing) in any kind
of metric
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 14 / 62
Exploring spacetimes via numerical computations: the geodesic code GYOTO
Gyoto code
Main developers: T. Paumard & F. Vincent
Integration of geodesics inKerr metric
Integration of geodesics inany numerically computed3+1 metric
Radiative transfer includedin optically thin media
Exploring spacetimes via numerical computations: the geodesic code GYOTO
3+1 geodesic integration in Gyoto code (1/2)
Numerical spacetime =⇒ (N, βi, γij ,Kij)
System to be integrated
dE
dt= E
(NKjkV
jV k − V j∂jN)
dXi
dt= NV i − βi
dV i
dt= NV j
[V i(∂j lnN −KjkV
k)
+ 2Kij − 3ΓijkV
k]− γij∂jN − V j∂jβi
Integration (backward) in time: Runge–Kutta algorithms of fourth to eighth order
Problem: the 3+1 quantities (N, βi, γij ,Kij) and their spatial derivatives have tobe known at any point along the geodesic and not only at the grid points issuedfrom the numerical relativity computation
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 19 / 62
Exploring spacetimes via numerical computations: the geodesic code GYOTO
3+1 geodesic integration in Gyoto code (2/2)
Solution within spectral methods: thanks to their spectral expansions, the fields(N, βi, γij ,Kij) are actually known at any point !
For instance, a scalar field, like N , is expanded as
N(t, r, θ, ϕ) =∑i,`,m
Ni`m(t)Ti(r)Ym` (θ, ϕ)
with
Ti: Chebyshev polynomial of degree i
Y m` : spherical harmonic of index (`,m)
Within spectral methods, the discretization does not occur on the values in thephysical space (no grid !) but on the finite number of coefficients Ni`m
The data are (Ni`m(tJ)) for a finite series of time steps (tJ)0≤J≤Jmax
=⇒ the values (Ni`m(t)) at an arbitrary time t are obtained by a third orderinterpolation from 4 neighbouring tJ ’s
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 20 / 62
Exploring spacetimes via numerical computations: the geodesic code GYOTO
Gyoto code
Computed images of a thin accretion disk around a Schwarzschild black hole
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 21 / 62
Exploring spacetimes via numerical computations: the geodesic code GYOTO
Measuring the spin from the black hole silhouette
Ray-tracing in the Kerr metric (spin parameter a)
Accretion structure around Sgr A* modelled as a ion torus, derived from thepolish doughnut class [Abramowicz, Jaroszynski & Sikora (1978)]
Exploring spacetimes via symbolic computations: the SageManifolds project
Outline
1 Astrophysical motivation: we are about to see black holes!
2 Exploring spacetimes via numerical computations: the geodesic code GYOTO
3 Exploring spacetimes via symbolic computations: the SageManifolds project
4 Conclusion and perspectives
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 24 / 62
Exploring spacetimes via symbolic computations: the SageManifolds project
Software for differential geometry
Packages for general purpose computer algebra systems:
xAct free package for Mathematica [J.-M. Martin-Garcia]
Ricci free package for Mathematica [J. L. Lee]
MathTensor package for Mathematica [S. M. Christensen & L. Parker]
DifferentialGeometry included in Maple [I. M. Anderson & E. S. Cheb-Terrab]
Atlas 2 for Maple and Mathematica
· · ·
Standalone applications:
SHEEP, Classi, STensor, based on Lisp, developed in 1970’s and 1980’s (free)[R. d’Inverno, I. Frick, J. Aman, J. Skea, et al.]
Cadabra field theory (free) [K. Peeters]
SnapPy topology and geometry of 3-manifolds, based on Python (free) [M.
Culler, N. M. Dunfield & J. R. Weeks]
· · ·
cf. the complete list on http://www.xact.es/links.htmlEric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 25 / 62
Exploring spacetimes via symbolic computations: the SageManifolds project
Sage approach to computer mathematics
Sage relies on a Parent / Element scheme: each object x on which somecalculus is performed has a “parent”, which is another Sage object X representingthe set to which x belongs.The calculus rules on x are determined by the algebraic structure of X.Conversion rules prior to an operation, e.g. x+ y with x and y having differentparents, are defined at the level of the parents
Example
sage: x = 4 ; x.parent()
Integer Ring
sage: y = 4/3 ; y.parent()
Rational Field
sage: s = x + y ; s.parent()
Rational Field
sage: y.parent().has_coerce_map_from(x.parent())
True
This approach is similar to that of Magma and different from that ofMathematica, in which everything is a tree of symbols
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 28 / 62
Exploring spacetimes via symbolic computations: the SageManifolds project
Exploring spacetimes via symbolic computations: the SageManifolds project
Implementing coordinate charts
Given a manifold M of dimension n, a coordinate chart on an open subsetU ⊂M is implemented in SageManifolds via the class Chart, whose main data isa n-uple of Sage symbolic variables x, y, ..., each of them representing acoordinate
In general, more than one (regular) chart may be required to cover the entiremanifold:
Examples:
at least 2 charts are necessary to cover the circle S1, the sphere S2, and moregenerally the n-dimensional sphere Sn
at least 3 charts are necessary to cover the real projective plane RP2
In SageManifolds, an arbitrary number of charts can be introduced
To fully specify the manifold, one shall also provide the transition maps onoverlapping chart domains (SageManifolds class CoordChange)
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 32 / 62
Exploring spacetimes via symbolic computations: the SageManifolds project
Implementing coordinate charts
Given a manifold M of dimension n, a coordinate chart on an open subsetU ⊂M is implemented in SageManifolds via the class Chart, whose main data isa n-uple of Sage symbolic variables x, y, ..., each of them representing acoordinate
In general, more than one (regular) chart may be required to cover the entiremanifold:
Examples:
at least 2 charts are necessary to cover the circle S1, the sphere S2, and moregenerally the n-dimensional sphere Sn
at least 3 charts are necessary to cover the real projective plane RP2
In SageManifolds, an arbitrary number of charts can be introduced
To fully specify the manifold, one shall also provide the transition maps onoverlapping chart domains (SageManifolds class CoordChange)
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 32 / 62
Exploring spacetimes via symbolic computations: the SageManifolds project
Implementing coordinate charts
Given a manifold M of dimension n, a coordinate chart on an open subsetU ⊂M is implemented in SageManifolds via the class Chart, whose main data isa n-uple of Sage symbolic variables x, y, ..., each of them representing acoordinate
In general, more than one (regular) chart may be required to cover the entiremanifold:
Examples:
at least 2 charts are necessary to cover the circle S1, the sphere S2, and moregenerally the n-dimensional sphere Sn
at least 3 charts are necessary to cover the real projective plane RP2
In SageManifolds, an arbitrary number of charts can be introduced
To fully specify the manifold, one shall also provide the transition maps onoverlapping chart domains (SageManifolds class CoordChange)
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 32 / 62
Exploring spacetimes via symbolic computations: the SageManifolds project
Implementing scalar fields
A scalar field on manifold M is a smooth mapping
f : U ⊂M −→ Rp 7−→ f(p)
where U is an open subset of M
A scalar field maps points, not coordinates, to real numbers=⇒ an object f in the ScalarField class has different coordinaterepresentations in different charts defined on U .
The various coordinate representations F , F , ... of f are stored as a Pythondictionary whose keys are the charts C, C, ...:
f. express ={C : F, C : F , . . .
}with f( p︸︷︷︸
point
) = F ( x1, . . . , xn︸ ︷︷ ︸coord. of pin chart C
) = F ( x1, . . . , xn︸ ︷︷ ︸coord. of pin chart C
) = . . .
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 33 / 62
Exploring spacetimes via symbolic computations: the SageManifolds project
Implementing scalar fields
A scalar field on manifold M is a smooth mapping
f : U ⊂M −→ Rp 7−→ f(p)
where U is an open subset of M
A scalar field maps points, not coordinates, to real numbers=⇒ an object f in the ScalarField class has different coordinaterepresentations in different charts defined on U .
The various coordinate representations F , F , ... of f are stored as a Pythondictionary whose keys are the charts C, C, ...:
f. express ={C : F, C : F , . . .
}with f( p︸︷︷︸
point
) = F ( x1, . . . , xn︸ ︷︷ ︸coord. of pin chart C
) = F ( x1, . . . , xn︸ ︷︷ ︸coord. of pin chart C
) = . . .
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 33 / 62
Exploring spacetimes via symbolic computations: the SageManifolds project
Implementing scalar fields
A scalar field on manifold M is a smooth mapping
f : U ⊂M −→ Rp 7−→ f(p)
where U is an open subset of M
A scalar field maps points, not coordinates, to real numbers=⇒ an object f in the ScalarField class has different coordinaterepresentations in different charts defined on U .
The various coordinate representations F , F , ... of f are stored as a Pythondictionary whose keys are the charts C, C, ...:
f. express ={C : F, C : F , . . .
}with f( p︸︷︷︸
point
) = F ( x1, . . . , xn︸ ︷︷ ︸coord. of pin chart C
) = F ( x1, . . . , xn︸ ︷︷ ︸coord. of pin chart C
) = . . .
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 33 / 62
Exploring spacetimes via symbolic computations: the SageManifolds project
The scalar field algebra
Given an open subset U ⊂M, the set C∞(U) of scalar fields defined on U hasnaturally the structure of a commutative algebra over R: it is clearly a vectorspace over R and it is endowed with a commutative ring structure by pointwisemultiplication:
∀f, g ∈ C∞(U), ∀p ∈ U, (f.g)(p) := f(p)g(p)
The algebra C∞(U) is implemented in SageManifolds via the classScalarFieldAlgebra.
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 34 / 62
Exploring spacetimes via symbolic computations: the SageManifolds project
Classes for scalar fields
Parent UniqueRepresentation
ScalarFieldAlgebra
Element: ScalarField
category: CommutativeAlgebras
SageManifolds class(differential part)
SageManifolds class(algebraic part)
native Sage class
ring: SR
CommutativeAlgebraElement
ScalarField
Parent: ScalarFieldAlgebra
ZeroScalarField
Parent: ScalarFieldAlgebra
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 35 / 62
Exploring spacetimes via symbolic computations: the SageManifolds project
Vector fields
Given an open subset U ⊂M, the set X (U) of smooth vector fields defined on Uhas naturally the structure of a module over the scalar field algebra C∞(U).
Modules vs. vector spaces
A module is ∼ vector space, except that it is based on a ring (here C∞(U))instead of a field (usually R or C in physics)
An importance difference: a vector space always has a basis, while a module doesnot necessarily have any
→ A module with a basis is called a free module
When X (U) is a free module, a basis is a vector frame (ea)1≤a≤n on U :
∀v ∈ X (U), v = vaea, with va ∈ C∞(U)
At a point p ∈ U , the above translates into an identity in the tangent vectorspace TpM:
v(p) = va(p) ea(p), with va(p) ∈ R
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 36 / 62
Exploring spacetimes via symbolic computations: the SageManifolds project
Vector fields
Given an open subset U ⊂M, the set X (U) of smooth vector fields defined on Uhas naturally the structure of a module over the scalar field algebra C∞(U).
Modules vs. vector spaces
A module is ∼ vector space, except that it is based on a ring (here C∞(U))instead of a field (usually R or C in physics)
An importance difference: a vector space always has a basis, while a module doesnot necessarily have any
→ A module with a basis is called a free module
When X (U) is a free module, a basis is a vector frame (ea)1≤a≤n on U :
∀v ∈ X (U), v = vaea, with va ∈ C∞(U)
At a point p ∈ U , the above translates into an identity in the tangent vectorspace TpM:
v(p) = va(p) ea(p), with va(p) ∈ R
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 36 / 62
Exploring spacetimes via symbolic computations: the SageManifolds project
Vector fields
Given an open subset U ⊂M, the set X (U) of smooth vector fields defined on Uhas naturally the structure of a module over the scalar field algebra C∞(U).
Modules vs. vector spaces
A module is ∼ vector space, except that it is based on a ring (here C∞(U))instead of a field (usually R or C in physics)
An importance difference: a vector space always has a basis, while a module doesnot necessarily have any
→ A module with a basis is called a free module
When X (U) is a free module, a basis is a vector frame (ea)1≤a≤n on U :
∀v ∈ X (U), v = vaea, with va ∈ C∞(U)
At a point p ∈ U , the above translates into an identity in the tangent vectorspace TpM:
v(p) = va(p) ea(p), with va(p) ∈ R
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 36 / 62
Exploring spacetimes via symbolic computations: the SageManifolds project
Vector fields
A manifold M that admits a global vector frame (or equivalently, such thatX (M) is a free module) is called a parallelizable manifold
Examples of parallelizable manifolds
Rn (global coordinate charts ⇒ global vector frames)
the circle S1 (NB: no global coordinate chart)
the torus T2 = S1 × S1
the 3-sphere S3 ' SU(2), as any Lie group
the 7-sphere S7
Examples of non-parallelizable manifolds
the sphere S2 (hairy ball theorem!) and any n-sphere Sn with n 6∈ {1, 3, 7}the real projective plane RP2
most manifolds...
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 37 / 62
Exploring spacetimes via symbolic computations: the SageManifolds project
Vector fields
A manifold M that admits a global vector frame (or equivalently, such thatX (M) is a free module) is called a parallelizable manifold
Examples of parallelizable manifolds
Rn (global coordinate charts ⇒ global vector frames)
the circle S1 (NB: no global coordinate chart)
the torus T2 = S1 × S1
the 3-sphere S3 ' SU(2), as any Lie group
the 7-sphere S7
Examples of non-parallelizable manifolds
the sphere S2 (hairy ball theorem!) and any n-sphere Sn with n 6∈ {1, 3, 7}the real projective plane RP2
most manifolds...
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 37 / 62
Exploring spacetimes via symbolic computations: the SageManifolds project
Vector fields
A manifold M that admits a global vector frame (or equivalently, such thatX (M) is a free module) is called a parallelizable manifold
Examples of parallelizable manifolds
Rn (global coordinate charts ⇒ global vector frames)
the circle S1 (NB: no global coordinate chart)
the torus T2 = S1 × S1
the 3-sphere S3 ' SU(2), as any Lie group
the 7-sphere S7
Examples of non-parallelizable manifolds
the sphere S2 (hairy ball theorem!) and any n-sphere Sn with n 6∈ {1, 3, 7}the real projective plane RP2
most manifolds...
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 37 / 62
Exploring spacetimes via symbolic computations: the SageManifolds project
Implementing vector fields
Ultimately, in SageManifolds, vector fields are to be described by theircomponents w.r.t. various vector frames.
If the manifold M is not parallelizable, one has to decompose it in parallelizableopen subsets Ui (1 ≤ i ≤ N) and consider restrictions of vector fields to thesedomains.
For each i, X (Ui) is a free module of rank n = dimM and is implemented inSageManifolds as an instance of VectorFieldFreeModule, which is a subclass ofFiniteRankFreeModule.
Each vector field v ∈ X (Ui) has different set of components (va)1≤a≤n indifferent vector frames (ea)1≤a≤n introduced on Ui. They are stored as a Pythondictionary whose keys are the vector frames:
v. components = {(e) : (va), (e) : (va), . . .}
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 38 / 62
Exploring spacetimes via symbolic computations: the SageManifolds project
Implementing vector fields
Ultimately, in SageManifolds, vector fields are to be described by theircomponents w.r.t. various vector frames.
If the manifold M is not parallelizable, one has to decompose it in parallelizableopen subsets Ui (1 ≤ i ≤ N) and consider restrictions of vector fields to thesedomains.
For each i, X (Ui) is a free module of rank n = dimM and is implemented inSageManifolds as an instance of VectorFieldFreeModule, which is a subclass ofFiniteRankFreeModule.
Each vector field v ∈ X (Ui) has different set of components (va)1≤a≤n indifferent vector frames (ea)1≤a≤n introduced on Ui. They are stored as a Pythondictionary whose keys are the vector frames:
v. components = {(e) : (va), (e) : (va), . . .}
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 38 / 62
Exploring spacetimes via symbolic computations: the SageManifolds project
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 39 / 62
Exploring spacetimes via symbolic computations: the SageManifolds project
Tensor field classes in SageManifolds
VectorField
Parent: VectorFieldModule
FreeModuleTensorTensorField
Parent: TensorFieldModule
TensorFieldParal
Parent: TensorFieldFreeModule
VectorFieldParalParent: VectorFieldFreeModule
FiniteRankFreeModuleElement
Parent: FiniteRankFreeModule
Parent: TensorFreeModule
Element
ModuleElement
Parent: Module
SageManifolds class(differential part)
SageManifolds class(algebraic part)
native Sage class
TangentVectorParent: TangentSpace
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 40 / 62
Exploring spacetimes via symbolic computations: the SageManifolds project
Tensor field storage
dictionary TensorField._restrictions
...
dictionary ScalarField._express
...
dictionary Components._comp
...
dictionary TensorFieldParal._components
...
...
Expression
Componentsframe 1:
FunctionChartchart 1:
ScalarField
(1,1) :
Componentsframe 2:
ScalarField
(1,2) :
FunctionChartchart 2:
Expression
TensorFieldParal
U1
U1
domain 1: TensorFieldParal
U2U2
domain 2:
TensorField
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 41 / 62
Exploring spacetimes via symbolic computations: the SageManifolds project
SageManifolds at work: the Mars-Simon tensor example
Definition [M. Mars, CQG 16, 2507 (1999)]
Given a 4-dimensional spacetime (M, g) endowed with a Killing vector field ξ, theMars-Simon tensor w.r.t. ξ is the type-(0,3) tensor S defined by
Sαβγ := 4Cµαν[β ξµξν σγ] + γα[β Cγ]ρµν ξρ Fµν
where
γαβ := λ gαβ + ξαξβ , with λ := −ξµξµ
Cαβµν := Cαβµν +i
2ερσµν Cαβρσ, with Cαβµν being the Weyl tensor and
εαβµν the Levi-Civita volume form
Fαβ := Fαβ + i ∗Fαβ , with Fαβ := ∇αξβ (Killing 2-form) and
∗Fαβ :=1
2εµναβFµν (Hodge dual of Fαβ)
σα := 2Fµαξµ (Ernst 1-form)
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 42 / 62
Exploring spacetimes via symbolic computations: the SageManifolds project
Mars-Simon tensor
The Mars-Simon tensor provides a nice characterization of Kerr spacetime:
Theorem (Mars, 1999)
If g satisfies the vacuum Einstein equation and (M, g) contains a stationaryasymptotically flat end M∞ such that ξ tends to a time translation at infinity inM∞ and the Komar mass of ξ in M∞ is non-zero, then
S = 0 ⇐⇒ (M, g) is locally isometric to a Kerr spacetime
Let us use SageManifolds...
...to check the ⇐ part of the theorem, namely that the Mars-Simon tensor isidentically zero in Kerr spacetime.
NB: what follows illustrates only certain features of SageManifolds; other ones,like the multi-chart and multi-frame capabilities on non-parallelizable manifolds,are not considered in this example. =⇒ More examples are provided athttp://sagemanifolds.obspm.fr/examples.html
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 43 / 62
Exploring spacetimes via symbolic computations: the SageManifolds project
Mars-Simon tensor
The Mars-Simon tensor provides a nice characterization of Kerr spacetime:
Theorem (Mars, 1999)
If g satisfies the vacuum Einstein equation and (M, g) contains a stationaryasymptotically flat end M∞ such that ξ tends to a time translation at infinity inM∞ and the Komar mass of ξ in M∞ is non-zero, then
S = 0 ⇐⇒ (M, g) is locally isometric to a Kerr spacetime
Let us use SageManifolds...
...to check the ⇐ part of the theorem, namely that the Mars-Simon tensor isidentically zero in Kerr spacetime.
NB: what follows illustrates only certain features of SageManifolds; other ones,like the multi-chart and multi-frame capabilities on non-parallelizable manifolds,are not considered in this example. =⇒ More examples are provided athttp://sagemanifolds.obspm.fr/examples.html
Eric Gourgoulhon (LUTH) Exploring black hole spacetimes with computers IRMA, Strasbourg, 18 Sept. 2014 43 / 62